Isaac Newton: The Last Sorcerer. Michael White

model of the universe and spent his final years under house arrest. He managed to avoid the punishment of the heretic only by denouncing his convictions and what he knew to be fundamental truth before the Inquisition in 1633, just nine years before Newton’s birth.

      But, once started, nothing could stop the flow of progress. By the early part of the seventeenth century the stage was set for another series of revolutionary advances in mathematics that would pave the way for the work of Newton and the triumph of change from a geocentric to a mechanistic and heliocentric viewpoint – from Aristotelian guesswork to empiricism, observational precision and mathematical rigour. The man responsible for this last development before Newton was to take up the baton and reach the law of gravity and the development of the calculus was the French philosopher René Descartes.

      Descartes’s great mathematical breakthrough was the realisation that the equation was not the only way in which mathematical terms could be related. During the 1630s he devised the idea of constructing coordinates to represent pairs of numbers relating to algebraic terms (usually x and y). These came to be known as Cartesian coordinates and opened up the vast range of possibilities offered by the drawing of graphs – lines and curves bordered by axes. The technical name for this branch of mathematics is analytical geometry, and it first appeared in an appendix called ‘Geometry’ tacked on to the end of Descartes’s Discourse on the Method, which was first published in 1637.

      Descartes’s technique galvanised the world of mathematics, and within a few years of publication the Discourse had influenced the work of mathematicians and astronomers throughout Europe. Such men as the English mathematicians William Wallace and Isaac Barrow, as well as natural philosophers and mathematicians on the Continent, led by Pierre de Fermat and Christiaan Huygens, used Descartes’s findings as a springboard for their own efforts, which began to focus on the properties of the curves that could be drawn using Cartesian coordinates.

      A simple example is the graph produced by plotting the distance travelled by a ball dropped from a high tower against the time for which it has fallen. Galileo had shown that the speed of a ball increases with time. If after one second the ball has fallen 16 feet, after two seconds 64 feet and after three seconds 144 feet, clearly it is accelerating. If these values are plotted on a graph with speed on the y-axis and time on the x-axis a curve is produced.

       Figure 1. The curve produced by plotting distance against time.

      Now it is comparatively easy to calculate properties for straight-line graphs. For example, the area under a straight line can be calculated by simple geometry known to the Babylonians, and the gradient of a straight line (or its steepness) can be found by dividing the change in the values along the y-axis by the corresponding values along the x-axis. So, if the distance-time graph had been a straight line, the gradient would have given us the speed of the ball (the change in

       Figure 2. Calculating the gradient and the area under a straight line.

      distance with time). But how can the properties of a curve be calculated?

      It was soon realised that one way to determine properties of curves, such as the one in our problem, was to imagine them as constantly shifting straight lines: if a straight line was drawn next to a curve and touched it at a particular point, this line could approximate the curve at that point. Mathematicians called this straight line a tangent, and found that they could treat a tangent like any other straight line – they could, for example, find its gradient and therefore work out a value for the speed of the ball at that particular point. But this was still an approximation – and a very limited one at that.

      Simple problems concerning objects travelling in circular motion had been studied by earlier generations of philosophers, especially Galileo, but by the 1660s astronomers weaned on Kepler’s work were becoming interested in mathematical models to describe the

       Figure 3. The tangent to the curve.

      new celestial mechanics – the mathematics of how the planets maintain their orbits around the Sun. They of course realised that the mathematics of curves could lead to a fuller understanding of planetary motion, but limited solutions such as those offered by drawing tangents were not accurate enough to correlate with increasingly sophisticated methods of gathering observational data. Although the mathematicians and astronomers of Europe were exploring methods of working with curves and some, such as Fermat and the great English polymath Christopher Wren, came to very limited solutions that worked in specific cases, there was a need for general solutions, or methods that could be applied to all situations. Newton gradually became aware of this as he studied the work of his predecessors while an undergraduate student at Cambridge during the early 1660s. By the middle of the decade all the elements were in place for a mathematician of genius to produce the required new mathematics. And, thanks to a series of unpredictable events, Newton was able to find the time and inspiration to do just that.

       Chapter 5 A Toe in the Water

       It is probably true quite generally that in the history of human thinking the most fruitful developments frequently take place at those points where two different lines of thought meet. These lines may have roots in quite different parts of human culture, in different times or different cultural environments or different religious traditions: hence if they actually meet, that is, if they are at least so much related to each other that a real interaction can take place, then one may hope that new and interesting developments may follow.

      WERNER HEISENBERG¹

      When, in the spring of 1669, the Trinity fellow Francis Aston was preparing to leave on a European tour, he wrote to his friend Isaac Newton asking for his advice on how best to conduct himself and what to look out for on his travels. This is surprising, since Newton had never travelled abroad and had only recently made his first trip to London. But it illustrates the high esteem in which Newton was held by his colleagues so early in his career, even in connection with matters outside his area of expertise. More significant still is Newton’s reply to Aston’s letter, for, as well as asking his friend to gather alchemical information for him and to attempt to track down the famous alchemist Giuseppe Francesco Borri, then living in Holland, Newton went on to offer a long list of dos and don’ts as though he were a seasoned globe-trotter. These included the recommendation:

      If you be affronted, it is better in a foreign country to pass it by in silence or with jest though with some dishonour than to endeavour revenge; for in the first case your credit is none the worse when you return into England or come into other company that have not heard of the quarrel, but in the second case you may bear the marks of the quarrel while you live, if you outlive it at all.²

      The reason for this easy confidence is that by the 1660s Newton had already adopted what one of his biographers has called ‘a Polonius-like pose’.³ Even as a boy he had been confident to the point of alienating others, but Newton the man, the twenty-six-year-old fellow of Trinity College, Cambridge, six months away from accepting the Lucasian chair, was already so accomplished that if he had done nothing further with his life he would still have found a significant place in the history of science.

      Although his genius was realised by only a handful of associates in Cambridge and he was totally unknown to the scientific community, by 1669 Isaac Newton was in fact the most advanced mathematician of his age, creator of the calculus as well as elucidator of the basic principle behind the inverse-square nature of gravity and the theory of the nature of colours. Within the space